Kepler's laws, Satellite Orbits, Launch Vehicles PSLV & GSLV

The terms like Kepler’s laws, satellite orbits, geosynchronous orbit, geostationary orbit, polar orbit, PSLV, GSLV, etc. keep on appearing in the news columns whenever there is a satellite launch.
So I thought it is better to keep all the related concepts in one place.

Kepler’s laws of planetary motion (applicable to satellites also)

Kepler’s First Law: The orbit of a planet is an ellipse with the Sun at one of the two foci.
Kepler’s Second Law: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
In simple words, the speed of the planet increases as it nears the sun and decreases as it recedes from the sun.

The varying orbital speed of the earth (in the figure, the orbit of the earth is exaggerated)

Kepler’s Third Law: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Hankwang, Wikipedia

Orbital period (T): time taken by a plant to complete one revolution around the sun.

Semi Major Axis (a₁ and a₂): half of the major axis of the ellipse.

T₁²/a₁³ = T₂²/a₂³

In simple terms, the distance of a planet from the sun determines the time it takes for that planet to revolve around the sun (farther the planet is, greater the orbital period).

Planet	Orbital Period (T) in years	Average Distance (R) in AU	T²/R³
Mercury	0.241	0.39	0.98
Venus	.615	0.72	1.01
Earth	1.00	1.00	1.00
Mars	1.88	1.52	1.01

Perigee and Apogee

Most satellites orbit the earth in elliptical patterns.
When a satellite is at its farthest point from the earth, it is at the apogee of the orbit.
When a satellite is at its closest point to the earth, it is at the perigee of the orbit.
In accordance with Kepler’s second law, the satellites are fastest at the perigee and slowest at the apogee.

Why satellites revolve rather than staying still in space?

There are two important forces acting on the satellite:

the gravitational force which will pull the satellite towards earth and
the centrifugal force (due to revolution) which counters the gravitational pull.

Source

Revolution causes centrifugal force (the object tends to move away from the centre).
Higher the speed of the revolving satellite (orbital velocity), higher the centrifugal force.
Thus, by varying the speed (orbital velocity) of the satellite, we can make the satellite

fall back to earth by decreasing the orbital velocity (centrifugal force < gravitational force)
stay in its orbit by adjusting the speed so that the centrifugal force balances the gravitational pull (centrifugal force = gravitational force). (Lower the orbit, higher should be the orbital velocity).
escape earth’s influence by keeping the orbital velocity above the required speed (centrifugal force > gravitational force).

Types of satellite orbits

Low Earth Orbit (LEO: 200-2000 km)

International Space Station (400 km), the Hubble Space Telescope (560 km) and some observation satellites are all rotating the earth in Low Earth Orbit.
LEO is high enough to significantly reduce the atmospheric drag yet close enough to observe the earth (remote sensing).
In LEO, the satellite’s orbital period is much smaller than the earth’s rotational period (24 hours).
That is, the satellites in LEO complete multiple revolutions in 24 hours (Lower the orbit, higher should be the speed).

Source

What is the speed required to keep a satellite in LEO?

The speed is dependent on the distance from the centre of the Earth.
At an altitude of 200 km, the required orbital velocity is a little more than 27,400 kmph.
In the case of the space shuttle, it orbits the Earth once every 90 minutes at an altitude of 466 km.

Advantages of LEO

Low Earth Orbit is used for things that we want to visit often, like the International Space Station, the Hubble Space Telescope and some satellites (usually spy satellites and other observation satellites).
This is convenient for installing new instruments, experiments, and return to earth in a relatively short time.

Disadvantages of LEO

Atmospheric drag will lead to more fuel consumption and constant speed adjustments.
A satellite traveling in LEO do not spend very long over any one part of the Earth at a given time.
Hence, satellites in LEO are not suitable for communication and weather observation and forecasting.

Solution

One solution is to put a satellite in a highly elliptical orbit (eccentric orbit ― non-geosynchronous).
The other is to place the satellite in a geosynchronous orbit.

Highly Elliptical Orbits

Kepler’s second law: an object in orbit about Earth moves much faster when it is close to Earth than when it is farther away.
Perigee is the closest point and apogee is the farthest.
If the orbit is very elliptical, the satellite will spend most of its time near apogee (the furthest point in its orbit) where it moves very slowly.
Thus, it can be above a specific location most of the time.

Disadvantages of Highly Elliptical Orbits

In a highly elliptical orbit, the satellite has long dwell time over one area, but at certain times when the satellite is on the high speed portion of the orbit, there is no coverage over the desired area.

Solution

We could have two satellites on similar orbits but timed to be on opposite sides at any given time.
In this way, there will always be one satellite over the desired coverage area at all times.

If we want continuous coverage over the entire planet at all times, such as the Global Positioning System (GPS satellites are in Medium Earth Orbit though), then we must have a constellation of satellites with orbits that are both different in location and time.
In this way, there is a satellite over every part of the Earth at any given time.

Satellite constellation (Source)

Geosynchronous Orbits (GSO)

Another solution to the dwell time problem is to have a satellite whose orbital period is equal to the period of rotation of the earth (24 hrs) (satellite’s revolution is in sync with the earth’s rotation).
In this case, the satellite cannot be too close to the Earth because it would not be going fast enough to counteract the pull of gravity.
Using Kepler’s third law it is determined that the satellite has to be placed approximately 36,000 km away from the surface of the Earth (~42,000 km from the centre of the Earth) in order to remain in a GSO orbit.
By positioning a satellite so that it has infinite dwell time over one spot on the Earth, we can constantly monitor the weather in one location, provide reliable telecommunications service, etc.
The downside of a GSO is that it is more expensive to put and maintain something that high up.

Geostationary Orbit or Geosynchronous Equatorial Orbit (GEO)

A geostationary orbit or geosynchronous equatorial orbit is a circular geosynchronous orbit above Earth’s equator and following the direction of Earth’s rotation.
Because the satellite stays right over the same spot all the time, this kind of orbit is called “geostationary.”

Source and Credits: GIS Geography

Geostationary vs Geosynchronous

Geostationary Orbit or Geosynchronous Equatorial Orbit (GEO)	Geosynchronous Orbit

They are both geosynchronous orbits (orbital period = 24 hours). Line of sight transmission
Orbital path is circular.	Orbit is an inclined circle or an inclined ellipse.
Orbital tilt is zero.	The orbital tilt is non-zero (inclined orbit)
An observer on the ground would not perceive the satellite as moving and would see it as a fixed point in the sky	A person on a point on Earth, will see a satellite in this orbit in the same place in the sky at the same time of the day, every day. Since the orbit has some inclination and/or eccentricity, the satellite would appear to describe a more or less distorted figure-eight in the sky and would rest above the same spots of the Earth’s surface once per day.
There are a limited number of positions available (traffic jam, interference of signals due to more satellites in the same orbit and risk of damage due to space debris) in this orbit due to safety and manoeuvring limits.	There are more orbital planes and positions available to satellites using this technique
Can receive signals with a simple antenna as the satellite is in relatively same position (DTH, VSAT services). (Parabolic antenna is used to nullify the effect of atmospheric distortions)	Requires a parabolic antenna as the satellite’s position slightly changes longitudinally.
Steering the antenna is not required.	It may sometimes require steering the antenna to achieve line of sight

Medium Earth Orbits (MEO: 2000-36,000 km)

Medium Earth Orbits (MEO) range in altitude from 2,000 kms up to the geosynchronous orbit at 36,000 km which includes part of the lower and all of the upper Van Allen radiation belts.
The Van Allen Radiation Belt is a region of high energy charged particles moving at speeds close to that of light encircling the Earth which can damage solar cells, circuits, and shorten the life of a satellite or spacecraft.
Practical orbits therefore avoid these regions.

Polar Orbits (PO)

Satellites in these orbits fly over the Earth from pole to pole in an orbit perpendicular to the equatorial plane.
This orbit is used in surface mapping and observation satellites since it allows the orbiting satellite to take advantage of the earth’s rotation below to observe the entire surface of the Earth as it passes below.
Pictures of the Earth’s surface in applications such as Google Earth come from satellites in polar orbits.

Sun-synchronous orbits (SSO)

Polar orbit and sun-synchronous orbits are low earth orbits.
Sun-synchronous orbit is a near polar orbit in which the satellite passes over any given point of the planet’s surface at the same local mean solar time.
When a satellite has a sun-synchronous orbit, it means that the satellite has a constant sun illumination.
Because of the consistent lighting, the satellites in sun-synchronous orbit are used for remote sensing applications (image the Earth’s surface in visible or infrared wavelengths) like imaging, spying, etc.

Parking Orbit

It is not always possible to launch a space vehicle directly into its desired orbit.
The launch site may be in an inconvenient location or the launch window may be very short.
In such cases the vehicle may be launched into a temporary orbit called a parking orbit.
The parking obit provides more options for realising the ultimate orbit.
For manned space missions the parking orbit provides an opportunity to recheck the systems.

Hofmann transfer orbit

The transfer orbit is the orbit used to break out of the parking orbit and break into the geosynchronous or geostationary orbit.

Geosynchronous transfer orbit (GTO)

A geosynchronous transfer orbit is a Hohmann transfer orbit — an elliptical orbit used to transfer between two orbits in the same plane — used to reach geosynchronous or geostationary orbit.

Escape velocity

Escape velocity is the minimum launch velocity (assuming the object is launched straight up) required for an object to escape earth’s gravitational pull (it doesn’t fall back to earth).
One condition is that once launched the object is not supplied with any additional energy nor hindered by external force (like atmospheric drag) other than earth’s gravity.
The escape velocity required for an object to escape earth’s gravitational pull is ~11.2 m/s (40,000+ kmph).
It is neither feasible (atmospheric friction will turn it into ash) nor desirable (cannot place satellites in desired orbit) to launch rockets at escape velocity.

India’s Satellite Launch Vehicles

Polar Satellite Launch Vehicle (PSLV)

PSLV is an indigenously-developed expendable launch system.

Expendable launch system used only once to carry a payload into space. E.g. PSLV, GSLV, etc.

Reusable launch system is intended to allow for recovery of the system for later reuse. E.g. NASA’s space shuttles, SpaceX Falcon 9 rocket (reusable first stage and expendable second stage), etc.

PSLV was developed in 1990s by ISRO to place satellites (mostly remote sensing satellites) in polar and near polar (e.g. sun-synchronous orbit) Lower Earth Orbits.
However, over the last decade, several PSLV missions were successful in sending satellites towards geosynchronous transfer orbit.
E.g. Chandrayaan-1 – 2008 and Mars Orbiter Mission or Mangalyaan – 2014 were launched using PSLV.
PSLV can fly in different configurations depending on the mass of its payload and the target orbit.
These configurations vary the number and type of solid rocket boosters attached to the rocket’s first stage, while the four core stages remain the same across all configurations.
PSLV’s first stage and third stage are solid-fuelled stages.
PSLV’s second stage and fourth stage are liquid-fuelled stages.
The second stage engine, Vikas, is a derivative of France’s Viking engine.
The PSLV-C (PSLV Core Alone) version of the rocket does not use additional boosters, while the PSLV-DL, PSLV-QL and PSLV-XL use two, four and six boosters respectively.

The Workhorse of India’s space program

PSLV earned its title ‘the Workhorse of ISRO’ through consistently delivering various satellites to Low Earth Orbits, particularly the IRS (Indian Remote Sensing) series of satellites.
PSLV Payload Capacity to SSO: 1,750 kg
PSLV Payload Capacity to Sub-GTO: 1,425 kg
In forty-seven launches to date, PSLV has achieved success forty-four times.
Despite the failure of its maiden flight, PSLV went on to record thirty-six consecutive successful launches from 1999 to 2017.
PSLVs were used to place the IRNSS satellite constellation (3 in GEO and 4 in GSO) in orbit.

Geosynchronous Satellite Launch Vehicle (GSLV)

GSLV is also an expendable launch system.
The GSLV project was initiated to launch geosynchronous satellites (most of them are heavy for PSLV).
GSLV uses solid rocket booster and the liquid-fuelled Vikas engine, similar to those in PSLV.
GSLV has solid-fuelled first stage, liquid-fuelled second stage and a cryogenic third stage.
A Cryogenic rocket stage is more efficient and provides more thrust.
However, the cryogenic stage is technically a very complex system due to its use of propellants (liquid oxygen ― minus 183 °C and liquid hydrogen ― minus 253 °C) at extremely low temperatures.
India had to develop cryogenic technology indigenously as the US objected to Russia’s involvement citing Missile Technology Control Regime (MTCR) May 1992.
A new agreement was signed with Russia for cryogenic stages with no technology transfer.
GSLV rockets using the Russian Cryogenic Stage (CS) are designated as the GSLV Mk I.
GSLV rockets using the indigenous Cryogenic Upper Stage (CUS) are designated the GSLV Mk II.
GSLV Payload Capacity to LEO: 5,000 kg
GSLV Payload Capacity to GTO: 2,500 kg
GSLV’s primary payloads are heavy communication satellites of INSAT class (about 2,500 kg) that operate from Geostationary orbits (36000 km) and hence are placed in Geosynchronous Transfer Orbits by GSLV.
The satellite in GTO is further raised to its final destination by firing its in-built on-board engines.

Geosynchronous Satellite Launch Vehicle Mark III (GSLV-III)

GSLV-III is designed to launch satellites into geostationary orbit and is intended as a launch vehicle for crewed missions under the Indian Human Spaceflight Programme.
The GSLV-III has a higher payload capacity than GSLV.
GSLV-III Payload Capacity to LEO: 8,000 kg
GSLV-III Payload Capacity to GTO : 4000 kg